Search Results (63)

This study introduces a novel inertial-type iteration algorithm based on the Normal S iteration for the class of almost contraction mappings in Banach spaces. Traditional fixed point iterations often suffer from slow convergence and high computational cost; to address these limitations, the proposed framework incorporates an adaptive inertial-type parameter. We establish strong convergence of the algorithm and derive explicit a posteriori error estimates under weak contractive conditions. In addition, we demonstrate the asymptotic equivalence of the NS inertial-type trajectories with the classical Normal S iteration, provide a comprehensive weak $w^2-$stability analysis, and obtain sharp upper bounds for the data dependence problem. The practical performance of the algorithm is evaluated in two distinct computational domains: image deblurring via wavelet-based ${\ell }_1$ regularization and the generation of complex fractal patterns, including Julia and Mandelbrot sets. Numerical results show that the proposed inertial-type iteration algorithm significantly outperforms existing methods—such as Picard, Mann, Ishikawa, and standard Normal S iterations—achieving faster convergence, higher PSNR values in image restoration, and more stable basins of attraction in fractal visualizations. These findings highlight the effectiveness and versatility of the NS inertial-type iteration algorithm approach for both theoretical analysis and real-world applications. Full article

This paper addresses the optimal management of photovoltaic (PV) systems and distribution static synchronous compensators (D-STATCOMs) in modern electrical distribution networks. A mixed-integer nonlinear programming (MINLP) model is formulated which co-optimizes device placement, sizing, and multi-period dispatch to minimize the total annualized system costs while satisfying AC power flow and operational constraints. To solve this challenging problem, a decomposition methodology is proposed, wherein the binary location decisions for the PVs and D-STATCOMs are treated as predefined inputs, upon the basis of site selections commonly reported in the literature. With the integer variables fixed, the problem is reduced to a continuous nonlinear programming (NLP) subproblem for optimal capacity sizing and operational scheduling, which is solved using the interior point optimizer (IPOPT) via the Julia/JuMP environment. The core contribution of this work lies in its comprehensive demonstration of the economic superiority of variable reactive power injection over conventional fixed compensation schemes. Through numerical validation on standard 33- and 69-bus test systems, it is shown that a variable D-STATCOM operation yields substantial and consistent economic gains. Compared to optimized fixed-injection solutions, variable injection provides additional annual savings averaging USD 120,516 (33-bus feeder) and USD 125,620 (69-bus grid), corresponding to a further 3.4% reduction in total costs. These benefits prove robust across different device location sets identified by various metaheuristic algorithms, and they scale effectively to larger network topologies. The results demonstrate that transitioning to variable power injection is not merely an incremental improvement but a fundamental advancement for achieving techno-economic optimality in distribution system planning. The proposed methodology provides utilities with a computationally efficient framework for determining near-optimal PV and D-STATCOM management strategies by first fixing deployment locations based on established planning insights and then rigorously optimizing sizing and dispatch, in order to maximize economic returns while ensuring reliable network operation. Full article

This study offers a computational framework that analyzes the escape characteristics of transcendental complex maps by utilizing the AK iteration scheme. The well-known polynomial map of the form $z^n+c$ is generalized to the form $z^n+\sin \left(z\right)+\log \left(c^m\right)$, with $m\ge 1$ and $c\in \mathbb{C}\backslash \left\{0\right\}$, allowing the creation of complex fractal structures. A precise escape criterion is developed for the AK iteration scheme, ensuring the numerical stability of the scheme when applied to the construction of the Mandelbrot set and the Julia set. In order to validate the effectiveness of the developed framework, a comparative analysis is performed between the AK iteration scheme and the CR iteration scheme, focusing on the first parametric case of the Mandelbrot set and the Julia set. The average escape time, average number of iterations, non-escaping area index, and fractal dimension are analyzed with respect to the two iteration schemes. The numerical results indicate that the fractal structure obtained by the AK iteration scheme is different from the fractal structure obtained by the CR iteration scheme, showing the effectiveness of the AK iteration scheme as a powerful tool in the study of complex systems. Full article

This paper investigates the fractal dynamical behavior of a discrete Caputo fractional-order hepatitis C virus model. First, we analyze the stability of the system by using spectral radius and design the fractional-order controller based on coordinate transformation. Then, a nonlinear coupling controller is constructed to achieve synchronization between two fractional-order models with different parameters and different fractional orders, and the synchronization is supported by rigorous mathematical proof. Numerical simulations are used to verify the effectiveness of control and synchronization. Full article

Shannon’s differential entropy for continuous variables suffers from a fundamental limitation: it is not invariant under scale transformations. This makes entropy values dependent on the choice of measurement units rather than reflecting intrinsic properties of distributions. While Jaynes proposed the limiting density of discrete points (LDDP) as a theoretical solution, a concrete method for computing the required invariant measure has been lacking. This paper establishes a rigorous connection between Kullback–Leibler divergence and the invariant measure, providing theoretical proofs of invariance under affine transformations and a practical computational method. We prove that entropy normalized by the median of k-nearest neighbor distances is invariant under affine transformations (Theorems 1 and 2). The non-negativity of the resulting entropy has been validated empirically across all tested distribution families, though a complete theoretical proof remains an open question. This approach extends naturally to multivariate settings, enabling scale-invariant mutual information estimation. We provide open-source implementations in Julia (EntropyInvariant.jl) and Python (entropy_invariant) and demonstrate their advantages over traditional approaches, particularly for variables with disparate scales. Full article

Fractal dimension is widely used as a quantitative descriptor of structural complexity in digital images. However, its practical implementation often involves methodological and computational trade-offs. The compression-based estimator provides an information-theoretic formulation that operates directly on grayscale images without mandatory binarization. Although the method is theoretically grounded and has been applied in real-world scenarios, its implementation-level behavior and computational characteristics have not been systematically analyzed under controlled conditions. To address this gap, this work presents a structured GPU-enabled validation framework for this estimator using synthetic Julia sets with known theoretical fractal dimensions. By focusing on their planar boundaries, which enable direct ground-truth comparison across multiple resolutions, numerical accuracy, statistical stability, and execution time are jointly evaluated across CPU and GPU implementations. Furthermore, additional experiments assess sensitivity to progressive Gaussian blur and exploratory behavior on grayscale textures from the Brodatz dataset, revealing that boundary-dominated fractals consistently yield dimensions between 1 and 2, whereas volumetric textures produce values greater than 2 without modifying the estimation framework. Performance profiling identifies distinct computational regimes and highlights a trade-off between robustness and execution time in the double-compression GPU configuration. This approach establishes a reproducible evaluation framework that supports the practical deployment of compression-based fractal dimension estimation in large-scale and time-constrained image analysis systems. Full article

The paper explored the fractal, nonlinear and chaotic dynamics between oil prices, inflation, economic growth and unemployment in Turkiye from 1960 to 2024 and examined how energy market volatility propagated through the macroeconomy via complex, regime-dependent mechanisms. It developed a chaotic regression method and employed entropy-based measures (Shannon, Rényi and Tsallis), Lyapunov exponents, Lorenz and Rössler attractors, Julia set diagnostics and the chaos Granger causality test (Hiemstra–Jones). By nesting entropy, chaos and causality within a unified framework, it contributed methodological innovations and practical insights to the energy–economy literature. The chaotic regression results revealed that oil price shocks generated asymmetric and nonlinear responses in inflation, unemployment and growth that were characterized by chaos and sensitivity to initial conditions and demonstrated that oil shocks act as catalysts for nonlinear propagation and fractal macroeconomic dynamics. Julia set results determined that unemployment can be explained by inflation fractal size. Hiemstra–Jones method determined unidirectional causality from oil to both inflation, economic growth and unemployment. According to the results, adopting nonlinear and chaos-based modeling approaches is essential to understand the macroeconomic consequences of energy shocks. For policymakers, the evidence determined that the costs of disinflation or inflation control are sensitive to energy market volatility. The paper contributed to the energy–economy-econometrics literature by integrating entropy, chaos and causality analyses into the oil price–macroeconomy nexus by offering both methodological innovations and practical insights. Full article

In this paper, a new class of fifth-order Chebyshev–Halley-type methods with a single parameter is proposed by using the polynomial interpolation method. The convergence order of the new method is proved. The dynamic behavior of the new method on quadratic polynomials $P\left(x\right)=(x-a)(x-b)$ is analyzed, the strange fixed points and the critical points of the operator are obtained, the corresponding parameter planes and dynamic planes are drawn, the stability and convergence of the iterative method are visualized, and some parameter values with good properties are selected. The fractal results of the new method corresponding to different parameters about polynomial $G\left(x\right)$ are plotted. Numerical results show that the new method has less computing and higher computational accuracy than the existing Chebyshev–Halley-type methods. The fractal results show the new method has good stability and convergence. The numerical results of different iteration methods are compared and agree with the results of dynamic analysis. Full article

The immiscibility of most polymers leads to poor interfacial adhesion in blends, a critical challenge that often limits the mechanical performance of polymer composites. This research introduces shape-forming elements (SFEs), a novel class of coextrusion dies designed to create additional geometric complexity and control over interfacial architecture. Specifically inspired by Julia Set and T-Square fractals, SFEs were simulated, prototyped, and found to be effective in coextrusion of different-colored polymer clays. The SFEs were employed to coextrude architected composites consisting of a liquid crystalline polymer (Vectra A950) and a cycloaliphatic polyamide (Trogamid CX7323). Mechanical testing revealed a strong positive correlation between the draw ratio and both the tensile modulus (adjusted R² = 0.94) and tensile stress at break (adjusted R² = 0.84). However, experimental cross-sections significantly differed from simulation results. These discrepancies were attributed to interfacial instabilities caused by material incompatibility between the two polymers and potential moisture-induced defects. This finding highlights critical challenges that arise during practical processing, emphasizing the importance of addressing polymer compatibility and moisture management to realize the full potential of SFEs in designing advanced polymer composites with targeted properties. Full article

We present a novel and completely deterministic method to model chaotic orbits in nonlinear discrete dynamics, taking the quadratic map as an example. This method is based on the resurgent analysis developed by Écalle to perform the resummation of divergent power series given by asymptotic expansions in linear differential equations with variable coefficients. To determine the long-term behavior of the dynamics, we calculate the zeros of a function representing the unstable manifold of the system using Newton’s method. The asymptotic expansion of the function is expressed as a kind of negative power series, which enables the computation with high accuracy. By use of the obtained zeros, we visualize the set of homoclinic points. This set corresponds to the Julia set in one-dimensional complex dynamical systems. The presented method is easily extendable to two-dimensional nonlinear dynamical systems such as Hénon maps. Full article

In this paper, a pattern for visualizing fractals, namely Julia and Mandelbrot sets for complex functions of the form $T\left(u\right)=u^a-\xi u^2+ru+sin{\rho }^{\sigma }\mathrm{for}\mathrm{all}u\in \mathbb{C}$ and $a\in \mathbb{N}\setminus \left\{1\right\}$, $\xi \in \mathbb{C}$, $r,\rho \in \mathbb{C}\setminus \left\{0\right\}$ are created using novel fast convergent iterative techniques. The new iteration scheme discussed in this study uses s-convexity and improves earlier approaches, including the Mann and Picard–Mann schemes. Further, the proposed approach is amplified by unique escape conditions that regulate the convergence behavior and generate Julia and Mandelbrot sets. This new technique allows greater versatility in fractal design, influencing the shape, size, and aesthetic structure of the designs created. By modifying various parameters in the suggested scheme, a significant number of visually interesting fractals can be generated and evaluated. Furthermore, we provide numerical examples and graphic demonstrations to demonstrate the efficiency of this novel technique. Full article

Due to the increasing complexity of industrial systems, fault detection hinders the continuity of productivity. Also, many methods in industrial systems whose complexity increases over time have a mechanism based on human intervention. Therefore, the development of intelligent systems in fault detection is critical.. Avoiding false alarms in detecting real faults is one of the goals of these systems. Modern technology has the potential to improve strategies for detecting faults related to machine components. In this study, a hybrid approach was applied on two different datasets for fault detection. First, in this hybrid approach, data is given as input to the artificial neural network. Then, predictions are obtained as a result of training using the ANN mechanism with the feed forward process. In the next step, the error value calculated between the actual values and the estimated values is transmitted to the feedback layers. IWO (Invasive Weed Optimization) optimization algorithm is used to calculate the weight values in this hybrid structure. However the IWO optimization algorithm is designed to be initialized with fractal-based weighting. By this process sequence, it is planned to increase the global search power without getting stuck in local minima. Additionally, fractal-based initialization is an important part of the optimization process as it keeps the overall success and stability within a certain framework. Finally, a testing process is carried out on two separate datasets supplied by the Kaggle platform to prove the model’s success in failure detection. Test results exceed 98%. This success indicates that it is a successful model with high generalization ability. Full article

This paper introduces novel, non-classical Julia and Mandelbrot sets using the Jungck–Noor iterative method with s-convexity, and derives an escape criterion for higher-order complex polynomials of the form $z^n+\Im z^3-\Re z+\omega$, where $n\ge 4$ and $\Im ,\Re ,\omega \in \mathbb{C}$. The proposed method advances existing algorithms, enabling the visualization of intricate fractal patterns as Julia and Mandelbrot sets with enhanced complexity. Through graphical representations, we illustrate how parameter variations influence the color, size, and shape of the resulting images, producing visually striking and aesthetically appealing fractals. Furthermore, we explore the dynamic behavior of these sets under fixed input parameters while varying the degree n. The presented results, both methodologically and visually, offer new insights into fractal geometry and inspire further research. Full article

In this paper, we introduce a generalised formulation of the logistic map extended to the complex plane and correspondingly redefine the classical Mandelbrot and Julia sets within this broader framework. Central to our approach is the development of an escape criterion based on the Picard orbit, which underpins the escape-time algorithms employed for graphical approximations of these sets. We analyse the structural and dynamical properties of the resulting Mandelbrot and Julia sets, emphasising their inherent symmetries through detailed visualisations. Furthermore, we examine how variations in a key parameter of the generalised map affect two critical numerical metrics: the average escape time and the non-escaping area index. Our computational study reveals that, particularly for Julia sets, these dependencies are characterised by intricate, highly non-linear behaviour—highlighting the profound complexity and sensitivity of the system under this generalised mapping. Full article

This article investigates and analyzes the diverse patterns of Julia sets generated by new classes of generalized exponential and sine rational functions. Using a generalized viscosity approximation-type iterative method, we derive escape criteria to visualize the Julia sets of these functions. This approach enhances existing algorithms, enabling the visualization of intricate fractal patterns as Julia sets. We graphically illustrate the variations in size and shape of the images as the iteration parameters change. The new fractals obtained are visually appealing and attractive. Moreover, we observe fascinating behavior in Julia sets when certain input parameters are fixed, while the values of n and m vary. We believe the conclusions of this study will inspire and motivate researchers and enthusiasts with a strong interest in fractal geometry. Full article